Subgroups — #25123213

Order 11008.

600 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19135536	C₂	1	1	yes	maximal, characteristic
2	#19135532	C₂	1	2	yes	maximal
3	#19135532	C₂	1	1	yes	maximal, characteristic
4	#19135161	C₂	1	2	yes	maximal
5	#19135154	C₂	1	2	yes	maximal
6	#19135154	C₂	1	2	yes	maximal
7	#19135154	C₂	1	2	yes	maximal
8	#19134740	C₂	1	2	yes	maximal
9	C₄₃×#2399	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×#13063236	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
16	#14582654	C₂²	1	2	yes	—
17	#14582654	C₂²	1	2	yes	—
18	#14582654	C₂²	1	2	yes	—
19	#14582654	C₂²	1	2	yes	—
20	#14582654	C₂²	1	2	yes	—
21	#14582654	C₂²	1	2	yes	—
22	#14582654	C₂²	1	2	yes	—
23	#14582654	C₂²	1	2	yes	—
24	#14582654	C₂²	1	2	yes	—
25	#14582654	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#398	C₄	1	2	yes	—
29	C₄₃×#398	C₄	1	1	yes	characteristic
30	C₄₃×#398	C₄	1	1	yes	characteristic
31	C₄₃×#395	C₄	1	1	yes	characteristic
32	C₄₃×#395	C₄	1	1	yes	characteristic
33	C₄₃×#395	C₄	1	2	yes	—
34	C₄×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
36	C₄₃×#376	C₂²	1	1	yes	characteristic
37	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
38	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
39	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
40	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
41	C₄⋊(C₄₃⋊C₈)	—	4	1	no	—
42	C₄⋊(C₄₃⋊C₈)	—	4	1	no	—
43	C₄⋊(C₄₃⋊C₈)	—	4	2	no	—
44	#13063236	—	4	1	no	—
45	C₄×C₄₃⋊C₈	—	4	2	no	—
46	C₄×C₄₃⋊C₈	—	4	1	no	—
47	C₂²×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
49	C₄×Q₈×C₄₃	—	4	1	no	—
50	C₄×Q₈×C₄₃	—	4	1	no	—
51	C₄×Q₈×C₄₃	—	4	1	no	—
52	C₄×Q₈×C₄₃	—	4	1	no	—
53	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
64	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
67	#95×C₄₃	C₂×C₄	1	1	yes	characteristic
68	#95×C₄₃	C₂×C₄	1	2	yes	—
69	#95×C₄₃	C₂×C₄	1	1	yes	characteristic
70	#95×C₄₃	C₂×C₄	1	2	yes	—
71	#95×C₄₃	C₂×C₄	1	2	yes	—
72	#95×C₄₃	C₂×C₄	1	2	yes	—
73	#95×C₄₃	C₂×C₄	1	2	yes	—
74	C₂×Q₈×C₄₃	—	2	4	no	—
75	C₂×Q₈×C₄₃	—	2	2	no	—
76	C₂×Q₈×C₄₃	—	4	1	no	—
77	C₂×Q₈×C₄₃	—	4	1	no	—
78	C₂×Q₈×C₄₃	—	4	1	no	—
79	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₄₃	—	2	4	no	—
91	C₄⋊C₄×C₄₃	—	2	4	no	—
92	C₄⋊C₄×C₄₃	—	2	4	no	—
93	C₄⋊C₄×C₄₃	—	4	2	no	—
94	C₄⋊C₄×C₄₃	—	4	1	no	—
95	C₄⋊C₄×C₄₃	—	4	2	no	—
96	C₄⋊C₄×C₄₃	—	4	1	no	—
97	C₄⋊C₄×C₄₃	—	4	1	no	—
98	C₄⋊C₄×C₄₃	—	4	1	no	—
99	C₄²×C₄₃	—	2	1	no	—
100	C₄²×C₄₃	—	2	1	no	—
101	C₄²×C₄₃	—	2	1	no	—
102	C₄²×C₄₃	—	2	2	no	—
103	C₄²×C₄₃	—	2	1	no	—
104	C₄²×C₄₃	—	2	1	no	—
105	C₄²×C₄₃	—	2	2	no	—
106	C₄²×C₄₃	—	2	1	no	—
107	C₄²×C₄₃	—	2	2	no	—
108	C₄²×C₄₃	—	2	2	no	—
109	C₂×C₄₃⋊C₈	—	2	2	no	—
110	C₂×C₄₃⋊C₈	—	2	2	no	—
111	C₂×C₄₃⋊C₈	—	2	2	no	—
112	C₂×C₄₃⋊C₈	—	2	2	no	—
113	C₂×C₄₃⋊C₈	—	2	2	no	—
114	C₂×C₄₃⋊C₈	—	2	2	no	—
115	C₂×C₄₃⋊C₈	—	2	4	no	—
116	C₂×C₄₃⋊C₈	—	2	4	no	—
117	C₂×C₄₃⋊C₈	—	2	2	no	—
118	C₂×C₄₃⋊C₈	—	2	2	no	—
119	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
120	Q₈×C₄₃	—	4	4	no	—
121	Q₈×C₄₃	—	4	4	no	—
122	C₂×C₄×C₄₃	—	2	1	no	—
123	C₂×C₄×C₄₃	—	2	2	no	—
124	C₂×C₄×C₄₃	—	2	2	no	—
125	C₂×C₄×C₄₃	—	2	1	no	—
126	C₂×C₄×C₄₃	—	2	1	no	—
127	C₂×C₄×C₄₃	—	2	1	no	—
128	C₂×C₄×C₄₃	—	2	2	no	—
129	C₂×C₄×C₄₃	—	2	1	no	—
130	C₂×C₄×C₄₃	—	2	1	no	—
131	C₂×C₄×C₄₃	—	2	1	no	—
132	C₂×C₄×C₄₃	—	2	2	no	—
133	C₂×C₄×C₄₃	—	2	2	no	—
134	C₂×C₄×C₄₃	—	2	2	no	—
135	C₂×C₄×C₄₃	—	2	2	no	—
136	C₂×C₄×C₄₃	—	2	2	no	—
137	C₂×C₄×C₄₃	—	2	2	no	—
138	C₂×C₄×C₄₃	—	2	2	no	—
139	C₂×C₄×C₄₃	—	2	2	no	—
140	C₂×C₄×C₄₃	—	2	1	no	—
141	C₂×C₄×C₄₃	—	2	1	no	—
142	C₂×C₄×C₄₃	—	2	2	no	—
143	C₂×C₄×C₄₃	—	2	1	no	—
144	C₂×C₄×C₄₃	—	2	2	no	—
145	C₂×C₄×C₄₃	—	2	2	no	—
146	C₂×C₄×C₄₃	—	2	1	no	—
147	C₂×C₄×C₄₃	#131	1	2	yes	—
148	C₂×C₄×C₄₃	#131	1	4	yes	—
149	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
151	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
152	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
153	C₄₃⋊C₈	—	4	2	no	—
154	C₄₃⋊C₈	—	4	2	no	—
155	C₄₃⋊C₈	—	4	2	no	—
156	C₄₃⋊C₈	—	4	2	no	—
157	#23367	—	43	1	no	maximal, Sylow
158	C₂²×C₄₃	#518	1	1	yes	characteristic
159	C₂²×C₄₃	#519	1	2	yes	—
160	C₂²×C₄₃	C₂×#131	1	1	yes	characteristic
161	C₂²×C₄₃	C₂×#131	1	2	yes	—
162	C₂²×C₄₃	#568	1	1	yes	characteristic
163	C₄×C₄₃	—	2	2	no	—
164	C₄×C₄₃	—	2	4	no	—
165	C₄×C₄₃	—	4	2	no	—
166	C₄×C₄₃	—	4	1	no	—
167	C₄×C₄₃	—	4	1	no	—
168	C₄×C₄₃	—	4	1	no	—
169	C₄×C₄₃	—	4	1	no	—
170	C₄×C₄₃	—	4	1	no	—
171	C₄×C₄₃	—	4	2	no	—
172	C₄×C₄₃	—	4	2	no	—
173	C₄×C₄₃	#563	1	1	yes	characteristic
174	C₄×C₄₃	#563	1	1	yes	characteristic
175	C₄×C₄₃	#563	1	2	yes	—
176	#2399	D₄₃	1	1	yes	characteristic
177	#1985	—	43	1	no	—
178	#1984	—	43	2	no	—
179	#1984	—	43	1	no	—
180	#1917	—	43	2	no	—
181	#1916	—	43	2	no	—
182	#1916	—	43	2	no	—
183	#1916	—	43	2	no	—
184	#1840	—	43	2	no	—
185	C₂×C₄₃	#2381	1	1	yes	characteristic
186	C₂×C₄₃	#2976	1	1	yes	characteristic
187	C₂×C₄₃	#2977	1	1	yes	characteristic
188	C₂×C₄₃	#2985	1	2	yes	—
189	C₂×C₄₃	#2986	1	2	yes	—
190	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
191	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
192	C₂×C₄⋊C₈	—	43	1	no	—
193	C₂×C₄⋊C₈	—	43	1	no	—
194	C₂×C₄⋊C₈	—	43	2	no	—
195	C₂×#97	—	43	1	no	—
196	C₂×C₄×C₈	—	43	2	no	—
197	C₂×C₄×C₈	—	43	1	no	—
198	#398	Dic₄₃	1	2	yes	—
199	#398	Dic₄₃	1	1	yes	characteristic
200	#398	Dic₄₃	1	1	yes	characteristic
201	#395	Dic₄₃	1	1	yes	characteristic
202	#395	Dic₄₃	1	1	yes	characteristic
203	#395	Dic₄₃	1	2	yes	—
204	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
205	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
206	#376	C₂×D₄₃	1	1	yes	characteristic
207	#336	—	43	2	no	—
208	#336	—	43	2	no	—
209	#336	—	43	2	no	—
210	#336	—	43	2	no	—
211	#336	—	43	2	no	—
212	#336	—	43	2	no	—
213	#336	—	43	2	no	—
214	#336	—	43	2	no	—
215	#336	—	43	2	no	—
216	#336	—	43	2	no	—
217	C₄₃	#23367	1	1	yes	characteristic, complemented, Sylow
218	C₂²×Q₈	C₂×Dic₄₃	1	1	yes	characteristic
219	C₂²×Q₈	C₂×Dic₄₃	1	1	yes	characteristic
220	C₂²×C₈	—	43	2	no	—
221	C₂²×C₈	—	43	2	no	—
222	C₂²×C₈	—	43	2	no	—
223	C₂²×C₈	—	43	2	no	—
224	C₄×Q₈	—	4	1	no	—
225	C₄×Q₈	—	4	1	no	—
226	C₄×Q₈	—	4	1	no	—
227	C₄×Q₈	—	4	1	no	—
228	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
229	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
230	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
231	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
232	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
233	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
234	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
235	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
236	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
237	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
239	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
240	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
241	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
242	C₄⋊C₈	—	172	1	no	—
243	C₄⋊C₈	—	172	1	no	—
244	C₄⋊C₈	—	172	2	no	—
245	#97	—	172	1	no	—
246	C₄×C₈	—	172	1	no	—
247	C₄×C₈	—	172	2	no	—
248	#95	C₂×Dic₄₃	1	1	yes	characteristic
249	#95	C₂×Dic₄₃	1	2	yes	—
250	#95	C₂×Dic₄₃	1	1	yes	characteristic
251	#95	C₂×Dic₄₃	1	2	yes	—
252	#95	C₂×Dic₄₃	1	2	yes	—
253	#95	C₂×Dic₄₃	1	2	yes	—
254	#95	C₂×Dic₄₃	1	2	yes	—
255	C₂×Q₈	—	2	4	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	—	4	1	no	—
260	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
261	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
262	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
263	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
264	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
265	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
266	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
267	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
268	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
269	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
270	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
271	C₂×C₈	—	86	2	no	—
272	C₂×C₈	—	86	2	no	—
273	C₂×C₈	—	86	2	no	—
274	C₂×C₈	—	86	2	no	—
275	C₂×C₈	—	86	4	no	—
276	C₂×C₈	—	86	2	no	—
277	C₂×C₈	—	86	2	no	—
278	C₂×C₈	—	86	4	no	—
279	C₂×C₈	—	86	2	no	—
280	C₂×C₈	—	86	2	no	—
281	C₄⋊C₄	—	2	4	no	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	2	4	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	—	4	2	no	—
290	C₄²	—	2	1	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	2	1	no	—
297	C₄²	—	2	1	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	2	2	no	—
300	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
301	Q₈	—	4	4	no	—
302	Q₈	—	4	4	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	#13063332	1	2	yes	—
329	C₂×C₄	#13063332	1	4	yes	—
330	C₂×C₄	#13063370	1	1	yes	characteristic
331	C₂×C₄	#13063370	1	1	yes	characteristic
332	C₂×C₄	#13063373	1	2	yes	—
333	C₂×C₄	#13063373	1	2	yes	—
334	C₈	—	172	2	no	—
335	C₈	—	172	2	no	—
336	C₈	—	172	2	no	—
337	C₈	—	172	2	no	—
338	C₂²	#14583374	1	1	yes	characteristic
339	C₂²	#14583377	1	2	yes	—
340	C₂²	C₂×#13063332	1	2	yes	—
341	C₂²	C₂×#13063332	1	1	yes	characteristic
342	C₂²	#14583666	1	1	yes	characteristic
343	C₄	—	2	2	no	—
344	C₄	—	2	4	no	—
345	C₄	—	4	2	no	—
346	C₄	—	4	2	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	4	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	1	no	—
353	C₄	#14583648	1	1	yes	characteristic
354	C₄	#14583648	1	1	yes	characteristic
355	C₄	#14583648	1	2	yes	—
356	C₂	#19137776	1	1	yes	characteristic
357	C₂	#19143643	1	1	yes	characteristic
358	C₂	#19143651	1	1	yes	characteristic
359	C₂	#19143690	1	2	yes	—
360	C₂	#19143697	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator, socle, prime radical	C₂³×C₄₃	C₂³×C₄
Fitting	C₄₃×#2399	C₂
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#2399	D₄₃
hypocenter	C₄₃	#23367
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2399

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
217. C₄₃	#23367	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#19135536	2
#19135532	2
#19135161	2
#19135154	2
#19134740	2
C₄₃×#2399	2
#23367	43